La descente et la montée étendues: the spatially <Emphasis Type="Bold">d</Emphasis>-anisotropic and the spatio-temporal case

There is a great demand for statistical modeling of phenomena that evolve in both space and time, and thus, there is a growing literature on correlation function models for spatio-temporal processes. In particular, various properties of these correlation functions have been studied only for the merely spatial or temporal case, fact that constitutes a strong motivation for our work. The goal of this paper is to inspect some properties, obtained with respect to partial differentiation and integration, of stationary spatio-temporal correlation functions for which anisotropy is obtained through isotropy between components as in Fernández-Casal et al. (Stat Comput 13(2):127–136, 2003). We show that through partial differentiation and integration it is possible to obtain permissible spatio-temporal correlation functions in the space–time domain. Other new results regard specific classes of space–time correlations introduced in recent literature. A curious result arises by differentiating scale mixtures of Euclid’s hat. Work partially funded by grant MTM2004-06231 from the Spanish Ministery of Science and Education.     

A deformation/blurring-based spatio-temporal model

A key objective in spatio-temporal modeling consists of providing an appropriate representation of complexity in interactive spatio-temporal dynamics inherent to real phenomena. Propagated effect of dynamical spatial deformation provides a meaningful way to describe certain forms of heterogeneous behaviour; in particular, in relation to processes evolving in unstable media, or to account for the possible effect of covariates, to mention some significant interpretations. In this paper, the formulation of a discrete time and continuous space spatio-temporal interaction model with autoregressive dynamics, incorporating the effect of continuous deformation of the spatial support over time, is studied. Among other fields, this approach provides a suitable representation for a variety of geophysical and environmental applications. In particular, a vast family of heterogeneous models is generated from models which display homogeneity in the absence of deformation. Structural characteristics and variability properties, as well as self-consistency conditions for a limiting continuous-time approximation, are analyzed.     

Nonseparable stationary anisotropic space–time covariance functions

Obtaining new and flexible classes of nonseparable spatio-temporal covariances and variograms has resulted a key point of research in the last years. The goal of this paper is to introduce and develop new spatio-temporal covariance models taking into account the problem of spatial anisotropy. Recent literature has focused on the problem of full symmetry and the problem of anisotropy has been overcome. Here we propose a generalization of Gneiting’s (J Am Stat Assoc 97:590–600, 2002a) approach and obtain new classes of stationary nonseparable spatio-temporal covariance functions which are spatially anisotropic. The resulting structures are proved to have certain interesting mathematical properties, together with a considerable applicability.Work partially funded by grant MTM2004-06231 from the Spanish Ministry of Science and Education.     

Impedance-admittance regression analysis of magnetotelluric fields

The paper is devoted to the development of the theory of magnetotelluric field processing. A new method is proposed for complete consistent robust impedance-admittance regression estimation. This approach eliminates divergences between results obtained by independent solution of linear systems for the determination of impedance and admittance matrices and, moreover, is a good stabilizer of solution. Formulas for statistical estimates of results obtained by this method are derived. The theory developed in the paper is based on a linear algebraic approach. The regression problem is analyzed in a linear complex space with a nonrigidly specified metric. The metric of the solution space is formed concurrently with the solution of the problem on the basis of currently processed time series. All statistical estimations of results are performed in the metric obtained as a consequence of the solution and, in this sense, are optimal. The full regression model is effective for diagnosing the presence of electromagnetic fields that cannot be completely reduced to the plane wave approximation. the possibilities of the regression estimation are considered in detail on the basis of correlation analysis applied to sets of spectral components of electromagnetic fields.     

A stochastic framework for downscaling processes of spatial averages based on the property of spectral multiscaling and its statistical diagnosis on spatio-temporal rainfall fields

Spectral multi-scaling postulates a power-law type of scaling of spectral distribution functions of stationary processes of spatial averages, over nested and geometrically similar sub-regions of the spatial parameter space of a given spatio-temporal random field. Presently a new framework is formulated for down-scaling processes of spatial averages, following naturally from the postulate of spectral multi-scaling, and key ingredients required for its implementation are described. Moreover, results from an extensive diagnostic study are presented, seeking statistical evidence supportive of spectral multi-scaling. Such evidence emerges from two sources of data. One is a 13 year long historical record of radar observations of rainfall in southeastern UK (Chenies radar), with high spatial (2 km) and temporal (5 min) resolution. The other is an ensemble of rain rate fields simulated by a spatio-temporal random pulse model fitted to the historical data. The results are consistent between historical and simulated rainfall data, indicating frequency-dependent scaling relationships interpreted as evidence of spectral multi-scaling across a range of spatial scales.     

Kriging with external drift for functional data for air quality monitoring

Functional data featured by a spatial dependence structure occur in many environmental sciences when curves are observed, for example, along time or along depth. Recently, some methods allowing for the prediction of a curve at an unmonitored site have been developed. However, the existing methods do not allow to include in a model exogenous variables that, for example, bring meteorology information in modeling air pollutant concentrations. In order to introduce exogenous variables, potentially observed as curves as well, we propose to extend the so-called kriging with external drift—or regression kriging—to the case of functional data by means of a three-step procedure involving functional modeling for the trend and spatial interpolation of functional residuals. A cross-validation analysis allows to choose smoothing parameters and a preferable kriging predictor for the functional residuals. Our case study considers daily PM₁₀ concentrations measured from October 2005 to March 2006 by the monitoring network of Piemonte region (Italy), with the trend defined by meteorological time-varying covariates and orographical constant-in-time variables. The performance of the proposed methodology is evaluated by predicting PM₁₀ concentration curves on 10 validation sites, even with simulated realistic datasets on a larger number of spatial sites. In this application the proposed methodology represents an alternative to spatio-temporal modeling but it can be applied more generally to spatially dependent functional data whose domain is not a time interval.     

匀变速扩展的圆盘形断层的远场辐射理论（二）

本文第一部分已讨论在以下两条假设下的匀变速扩展的圆盘形断层的远场辐射理论: 1.破裂是从中心开始的;2.均匀位错分布（n=0）。 本部分将讨论更一般的情形,第一部分中的结果可以从本部分的普遍公式中作为特殊情形过渡得到。     

Testing the weak stationarity of a spatio-temporal point process

A common assumption in analyzing spatial and spatio-temporal point processes is stationarity, while in many real applications because of the environmental effects the stationarity condition is not often met. We propose two types of test statistics to test stationarity for spatio-temporal point processes, by adapting, Palahi, Pukkala & Mateu (2009) and by considering the square difference between observed and expected (under stationarity) intensities. We study the efficiency of the new statistics by simulated data, and we apply them to test the stationarity of real data.     

Spatio-temporal interpolation of precipitation during monsoon periods in Pakistan

Spatio-temporal estimation of precipitation over a region is essential to the modeling of hydrologic processes for water resources management. The changes of magnitude and space–time heterogeneity of rainfall observations make space–time estimation of precipitation a challenging task. In this paper we propose a Box–Cox transformed hierarchical Bayesian multivariate spatio-temporal interpolation method for the skewed response variable. The proposed method is applied to estimate space–time monthly precipitation in the monsoon periods during 1974–2000, and 27-year monthly average precipitation data are obtained from 51 stations in Pakistan. The results of transformed hierarchical Bayesian multivariate spatio-temporal interpolation are compared to those of non-transformed hierarchical Bayesian interpolation by using cross-validation. The software developed by [11] is used for Bayesian non-stationary multivariate space–time interpolation. It is observed that the transformed hierarchical Bayesian method provides more accuracy than the non-transformed hierarchical Bayesian method.     

Spatial autoregressive functional plug-in prediction of ocean surface temperature

This paper addresses the problem of spatial functional extrapolation in the framework of spatial autoregressive Hilbertian processes of order one (SARH(1) processes) introduced in Ruiz-Medina (J Muitivar Anal 102:292–305, 2011a). Moment-based estimators of the operators involved in the state equation of these processes are computed by projection into a suitable orthogonal basis. Specifically, the eigenfunction basis diagonalizing the autocovariance operator is considered. An estimation algorithm is designed for the implementation of the resulting SARH(1)-plug-in projection extrapolator from temporal curves irregularly distributed in space. Its performance is illustrated with a real-data example, where the problem of spatial functional extrapolation of ocean surface temperature profiles is addressed. This problem is crucial in the assessment of climate change anomalies. The data are collected from the public oceanographic bio-optical database: The World-wide Ocean Optics Database. Cross Validation (C.V.) procedures are applied for the evaluation of the estimation results derived.     

In spatio-temporal disease mapping models,identifiability constraints affect PQL and INLA results

Disease mapping studies the distribution of relative risks or rates in space and time, and typically relies on generalized linear mixed models (GLMMs) including fixed effects and spatial, temporal, and spatio-temporal random effects. These GLMMs are typically not identifiable and constraints are required to achieve sensible results. However, automatic specification of constraints can sometimes lead to misleading results. In particular, the penalized quasi-likelihood fitting technique automatically centers the random effects even when this is not necessary. In the Bayesian approach, the recently-introduced integrated nested Laplace approximations computing technique can also produce wrong results if constraints are not well-specified. In this paper the spatial, temporal, and spatio-temporal interaction random effects are reparameterized using the spectral decompositions of their precision matrices to establish the appropriate identifiability constraints. Breast cancer mortality data from Spain is used to illustrate the ideas.     

Nonlinear multivariate rainfall–stage model for large wetland systems

Wetland restoration is often measured by how close the spatial and temporal water level (stage) patterns are to the pre-drainage conditions. Driven by rainfall, such multivariate conditions are governed by nonstationary, nonlinear, and nonGaussian processes and are often simulated by physically based distributed models which are difficult to run in real time due to extensive data requirements. The objective of this study is to provide the wetland restorationists with a real time rainfall–stage modeling tool of simpler input structure and capability to recognize the wetland system complexity. A dynamic multivariate Nonlinear AutoRegressive network with eXogenous inputs (NARX) combined with Principal Component Analysis (PCA) was developed. An implementation procedure was proposed and an application to Florida Everglade’s wetland systems was presented. Inputs to the model are time lagged rainfall, evapotranspiration and previously simulated stages. Data locations, preliminary time lag selection, spatial and temporal nonstationarity are identified through exploratory data analysis. PCA was used to eliminate input variable interdependence and to reduce the problem dimensions by more than 90% while retaining more than 80% of the process variance. A structured approach to select optimal time lags and network parameters was provided. NARX model results were compared to those of the linear Multivariate AutoRegressive model with eXogenous inputs. While one step ahead prediction shows comparable results, recursive prediction by NARX is far more superior to that of the linear model. Also, NARX testing under drastically different climatic conditions from those used in the development demonstrates a very good and robust performance. Driven by net rainfall, NARX exhibited robust stage prediction with an overall Efficiency Coefficient of 88%, Mean Square Error less than 0.004 m², a standard error less than 0.06 m, a bias close to zero and normal probability plots show that the errors are close to normal distributions.     

欧拉梁动力反应初边值问题的微分求积解

动荷载作用下欧拉梁动响应的计算是一个初边值问题,通常很难得到解析解,传统数值方法一般是把空间和时间分别离散进行求解,计算相对复杂,效率也不高.针对分布动荷载作用下欧拉梁的振动偏微分方程,采用传统微分求积法,在空间和时间上同时进行离散;对于所有非0阶的初/边值条件,采用嵌入法在权系数计算中予以考虑.算例的数值结果与精确解的对比证明采用传统微分求积法处理此问题是可行的,而且是高效的.对于实际工程中的其他类似问题,该方法同样适用.     

Modal model identification of structures under unmeasured seismic excitations

Techniques developed for structural identification of a structural model are typically based on information regarding the response and the forcing actions. However, in some situations it can be necessary, or simply useful, to refer only to the measured responses. In this paper we describe a technique suitable for identifying the modal model of a spatial frame in the frequency domain when the seismic input is unknown both in time contents and direction. In some previous theoretical works we established that this identification problem has a unique solution when at least three time‐history responses are known. Here numerical techniques are developed which allow the evaluation of the modal quantities in practice. Numerical applications are carried out on plane and spatial framed structures by using a modal model which may be complete, including all the structure's modes, or incomplete, including only the lowest modes. In most cases the obtained results are satisfactory. Copyright © 2005 John Wiley & Sons, Ltd.     

On the theory of thin rotating jets: A quasi‐geostrophic time dependent model

The time‐dependent meandering in a thin baroclinic jet over bottom topography is discussed in the quasi‐geostrophic approximation. The motion of the axis of the jet is taken to be vertically coherent and the axis itself is defined as inextensible. The motion is examined from a frame of reference moving with the axis but fixed at an arbitrary longitude in terms of an open ocean spatial initial value problem. The velocities of the axis and of the jet are quasi‐geostrophic, and vorticity conservation for the first non‐geostrophic components constrains the evolution of the axis and gives a path equation. The spatial linearized stability problem is studied and the jet is found to be baroclinically unstable to path disturbances of sufficiently high frequency which amplify downstream. An exact solution is obtained to the nonlinear path equation over a flat bottom with no ß‐effect. The evolution of the path of these unstable meanders is such that the path closes itself and forms rings (at which point the analysis breaks down). It is proposed that the baroclinic jet processes studied here are fundamental to the dynamics of Gulf Stream meandering and isolated eddy production.     

Stochastic partial differential equations in groundwater hydrology

Many problems in hydraulics and hydrology are described by linear, time dependent partial differential equations, linearity being, of course, an assumption based on necessity.Solutions to such equations have been obtained in the past based purely on deterministic consideration. The derivation of such a solution requires that the initial conditions, the boundary conditions, and the parameters contained within the equations be stipulated in exact terms. It is obvious that the solution so derived is a function of these specified, values.There are at least four ways in which randomness enters the problem. i) the random initial value problem; ii) the random boundary value problem; iii) the random forcing problem when the non-homogeneous part becomes random and iv) the random parameter problem.Such randomness is inherent in the environment surrounding the system, the environment being endowed with a large number of degrees of freedom.This paper considers the problem of groundwater flow in a phreatic aquifer fed by rainfall. The goveming equations are linear second order partial differential equations. Explicit form solutions to this randomly forced equation have been derived in well defined regular boundaries. The paper also provides a derivation of low order moment equations. It contains a discussion on the parameter estimation problem for stochastic partial differential equations.     

Critical earthquake input power spectral density function models for engineering structures

The problem of determining optimal power spectral density models for earthquake excitation which satisfy constraints on total average power, zero crossing rate and which produce the highest response variance in a given linear system is considered. The solution to this problem is obtained using linear programming methods. The resulting solutions are shown to display a highly deterministic structure and, therefore, fail to capture the stochastic nature of the input. A modification to the definition of critical excitation is proposed which takes into account the entropy rate as a measure of uncertainty in the earthquake loads. The resulting problem is solved using calculus of variations and also within linear programming framework. Illustrative examples on specifying seismic inputs for a nuclear power plant and a tall earth dam are considered and the resulting solutions are shown to be realistic.     

Geospatial–temporal dependence among weekly precipitation extremes with applications to observations and climate model simulations in South America

A quantification of the spatio-temporal dependence among precipitation extremes is important for investigating the properties of intense storms as well as flood or flash-flood related hazards. Extreme value theory has been widely applied to the hydrologic sciences and hydraulic engineering. However, rigorous approaches to quantify dependence structures among extreme values in space and time have not been reported in the literature. Previous researchers have quantified the dependence among extreme values through the concept of (pairwise bivariate) tail dependence coefficients. For estimation of the tail dependence coefficients, we apply a recently developed method [Kuhn G. On dependence and extremes. PhD thesis (Advisor: C. Klüppelberg), Munich University of Technology, 2006] which utilized the multivariate tail dependence function of a subclass of elliptical copulas. This study extends the previous approach in the context of space and time by considering pairs of spatial grids in South America and quantifying the dependence among precipitation extremes based on the time series at each spatial grid. In addition, Kendall’s τ is used to estimate the pairwise copula correlation (for an elliptical copula) of precipitation between all grids in South America. The geospatial–temporal dependence measures are applied to precipitation observations from 1940 to 2005 as well as simulations from the Community Climate System Model version 3 (CCSM3) for 1940–2099. New insights are obtained regarding the spatio-temporal dependence structures for precipitation over South America both with regard to correlation as well as tail dependence.     

Spatio-temporal epidemic type aftershock sequence model for Tangshan aftershock sequence

Shallow earthquakes usually show obvious spatio-temporal clustering patterns. In this study, several spatio-temporal point process models are applied to investigate the clustering characteristics of the well-known Tangshan sequence based on classical empirical laws and a few assumptions. The relative fit of competing models is compared by Akaike Information Criterion. The spatial clustering pattern is well characterized by the model which gives the best fit to the data. A simulated aftershock sequence is generated by thinning algorithm and compared with the real seismicity.     

Analysis of floodplain inundation using 2D nonlinear diffusive wave equation solved with splitting technique

In the paper a solution of two-dimensional (2D) nonlinear diffusive wave equation in a partially dry and wet domain is considered. The splitting technique which allows to reduce 2D problem into the sequence of one-dimensional (1D) problems is applied. The obtained 1D equations with regard to x and y are spatially discretized using the modified finite element method with the linear shape functions. The applied modification referring to the procedure of spatial integration leads to a more general algorithm involving a weighting parameter. Time integration is carried out using a two-level difference scheme with the weighting parameter as well. The resulting tri-diagonal systems of nonlinear algebraic equations are solved using the Picard iterative method. For particular sets of the weighting parameters, the proposed method takes the form of a standard finite element method and various schemes of the finite difference method. On the other hand, for the linear version of the governing equation, the proper values of the weighting parameters ensure an approximation of 3rd order. Since the diffusive wave equation can be solved no matter whether the area is dry or wet, the numerical computations can be carried out over entire domain of solution without distinguishing a current position of the shoreline which is obtained as a result of solution.